Which is funny, because ice-boxes used to have a large chunk of ice at the top. I remember vaguely seeing the iceman come(th) as a very small child. My husband remembers the ice house where they got ice for the Coca Cola truck cooler. (His father's employer.)

However, I (northeast coast to OH and now CA, b 1943) would never use that term now. It's the refrigerator, possibly the fridge. My husband, from NC and Georgia, b 1940,

Some people would say Frigidaire like I still say Kleenex.
Where are you from?

Why is "I have to put the milk into the fridge" not accepted?
When I learned english in school we were told to use the term "into"when we put something inside or go inside.
Is that not common english anymore?
I reported it anyway

It's actually not common to say into in that situation, but it makes sense with your reasoning and should in accepted.
However you have now discovered another English exception. I don't know where the division lies. I'm going into the house can also be I'm going in the house.
I asked my husband for a sentence with into and he agreed that in could also have been used. He thought maybe it implies more intention.
into can also be used as in I'm really into my hobby.
So maybe into is dying out....but still correct!

Report it, it should be. In can be used for stative situations ( The box is in the bag), in which case you can usually substitute inside. It can also be used for active situations ( Put the box in the bag), in which case into can usually be substituted. Exactly where the divisions between in/into and in/inside gets made is probably a matter of dialect, idiolect, context, or a mixture of all three. Always using in won't necessarily mark you as a non-native English speaker.

"tener" by itself means "to have (something)" and would be followed by a noun.

I have the milk=Tengo la leche.

But "I have put the milk..." ("put" here is a past participle, like "placed, seen"...)="He puesto la leche" ...(not with tener, but a form of haber+past participle, — which is otherwise only used as "There is/are"=Hay leche en la nevera)

And then there's "I have to" (i.e. I need to + verb infinitive, where the "to" isn't the infinitive "to", but part of the "have to" expression) "I have to - put the milk..."= "Tengo que - poner la leche..."

So "have" has many meanings in English, and 2 use tener in Spanish in different ways, and one uses "haber" which also has a different use.

"I have put milk in the fridge" is a general statement implying that in the past you have used the refrigerator to store milk. This is as opposed to putting some specific bottle of milk in the refrigerator, like "Where did you put the milk?", "I put the milk in the fridge".

I'm curious to know where?
This word seems to have many meanings, depending on geography, and otherwise, a refrigerator can also be 'refrigerador' or 'refrigeradora' as well as 'nevera' in different locations.
And what other words are used, and where, for freezer?